Preheat the oven to 180 degrees C (355 degrees F).
Spread butter over one side of all the bread slices. Push a slice of bread, buttered-side down, inside each cup of a large muffin tin to make bread cups.
In a large mixing bowl, beat the egg whites until light and foamy. Add the yolks and beat for a further minute. Add the Cheddar, ham, chives, parsley, paprika, mushrooms, garlic and red pepper to the egg bowl and gently fold in until incorporated. Season with salt and pepper.
Spoon the egg mixture evenly into the bread cups and bake until the bread is lightly golden, about 20 minutes.